Alice thinks about a natural number in her mind. Bob tries to find that number by asking him the following 10 questions: Is it divisible by 1? Is it divisible by 2? Is it divisible by 3? ... Is it divisible by 9? Is it divisible by 10? Alice's answer to all questions except one was "yes". When she answers "no", she adds that "the greatest common factor of the number I have in mind and the divisor in the question you asked is 1”. According to this information, to which question did Alice answer "no"?

Find all the real number triples $(x, y, z)$ satisfying the following system of inequalities under the condition $0 < x, y, z < \sqrt{2}$: $$y\sqrt{4-x^2y^2}\ge \frac{2}{\sqrt{xz}}$$$$x\sqrt{4-x^2z^2}\ge \frac{2}{\sqrt{yz}}$$$$z\sqrt{4-y^2z^2}\ge \frac{2}{\sqrt{xy}}$$.

Find all the natural numbers $a, b, c$ satisfying the following equation: $$a^{12} + 3^b = 1788^c$$.

A $9 \times 10$ board is divided into $90$ unit cells. There are certain rules for moving a non-standard chess queen from one square to another: The queen can only move along the column or row it is in each step. For any natural number $n$, if $x$ cells move made in $(2n-1)$th step, then $9-x$ cells move will be done in $(2n)$th step. The last cell it stops at during these steps is considered the visited cell. Is it possible for the queen to move from any square on the board and return to the square where it started after visiting all the squares of the board exactly once? Note: At each step, the queen chooses the right, left, up, and down direction within the above condition can choose.

In a scalene triangle $ABC$, the points $E$ and $F$ are the foot of altitudes drawn from $B$ and $C$, respectively. The points $X$ and $Y$ are the reflections of the vertices $B$ and $C$ to the line $EF$, respectively. Let the circumcircles of the $\triangle ABC$ and $\triangle AEF$ intersect at $T$ for the second time. Show that the four points $A, X, Y, T$ lie on a single circle.